Since, we will be dealing with concentration phenomenon for Harris chains which are not contractive w.r.t trivial euclidean metric, more general inequality linking Wasserstein metric and relative entropy is required that allows for the flexibility of distance like \(d_{\beta ^*}\) . Shortly after [1]} came up with weighted T-E inequalities, in a totally independent setting of interest [2]} proved convergence of Harris chain in \(\mathcal {W}_{d_{\beta ^*}}\) , when used together these ideas allowed us to extend concentration phenomenon to nonlinear random dynamical systems. Since Wasserstein metric is always upper bounded by a weighted total variation distance, in this setting weight is \(1+\beta ^* V(\cdot )\)  and we have the following wighted T-E inequality:
\(& \nonumber \mathcal {W}_{d_{\beta ^*}} (\mathcal {P}(x. \cdot ),\nu ) \le \Vert (1+\beta ^* V(y) ) (\mathcal {P}(x. \cdot ) -\nu )(dy) \Vert _{tv} \le \\  \le & 2\underbrace{\inf _{\alpha >0}\bigg [\frac{1}{2 \alpha } \bigg (1+\log \int e^{\alpha (1+\beta ^* V(y))^2} \mathcal {P}(x, dy) \bigg ) \bigg ]^{\frac{1}{2}}}_{C(\alpha ,\beta ^*, x)} \times \sqrt{Ent(\nu || \mathcal {P}(x, \cdot ))}.\) 
